Three first nights
First night in Hakata: shinkansen and the airport line both use Hakata Station, about 5 minutes from the airport. Shopping in Tenjin: the underground and Nishitetsu are a walk, Tenjin to Nishitetsu Fukuoka is a 6-minute next stop. Yatai in Nakasu: Nakasu-Kawabata exits onto the river. Do not sleep in Hakata and walk back for a stall.
Whether to sleep out
Momochi is for event days: sleep there only with a dome ticket, otherwise return to Tenjin or Hakata. Dazaifu follows the Nishitetsu last train: be back at Futsukaichi for Tenjin before it, and do not time it on the subway. Itoshima needs a car or a bus: the Chikuhi Line runs through to Maebaru, and the coast is another bus. Without a car or a bus you can catch, do not base there. Yanagawa is whether you can still reach Tenjin after the punt: the station to the boat is a 16-minute next stop, and after the boat you must be back at Nishitetsu Yanagawa before the last train. If not, stay the night and do not push on to Dazaifu. Do not do Dazaifu and Yanagawa the same day.
Where to be before the last train
No last-train minute is locked in, so none is written. Dazaifu: be at Futsukaichi or Tenjin before the last train. Itoshima: be at Chikuzen-Maebaru before the last bus, then Meinohama. Yanagawa: be at Nishitetsu Yanagawa before the last train to Tenjin. Uminonakamichi: be at Kashii before the last train, then Hakata. Subway tap caps at 640 yen a day. Nishitetsu and JR are outside it. The dossier has no rates and no booking.